COLLEGE OF ARTS AND SCIENCES
DEPARTMENT OF MATHEMATICS
MATHEMATICS IN THE
MODERN WORLD
MODULE 1
________________________
Name of Student
_______________________________________
Course and Year
ABOUT THE COURSE
Course Number: Math 1
Course Title: Mathematics in the Modern World
Course Description: This course deals with the nature of mathematics, appreciation of its practical
and intellectual and aesthetic dimensions, and application of mathematical tools in
daily life.
The course begins with an introduction to the nature of mathematics as an
exploration of pattern (in nature and the environment) and as an application of
inductive and deductive reasoning. By exploring these topics, students are
encouraged to go beyond the typical understanding of mathematics as merely a set
of formulas but as a source of aesthetics in patterns of nature, for example, and a
rich language in itself (and of science) governed by logic and reasoning.
The course then proceeds to survey ways in which mathematics provide a tool for
understanding and dealing with various aspects of present – day living, such as
managing personal finances, making social choices, appreciating geometric designs,
understanding codes used in data transmission and security, and dividing limited
resources fairly. These aspects will provide opportunities for actually doing
mathematics in a broad range of exercises.
Pre-requisite: None
Credit Units: 3 Units
Credit Hours: 54 hours per semester or 3 hours per week for 18 weeks
Course Outline
Section I. The Nature of Mathematics
1. Mathematics in our World
 Patterns and Numbers in Nature and the World;
a. The snowflakes and honeycomb;
b. Tiger’s stripes and hyena’s spots;
c. The sunflower;
d. The snail’s shell;
e. Flower petals;
f. The world’s population, the weather, etc.
No. of Hours
9 hours
References:
CHED Memorandum Order No. 20, series of 2013. (4 July 2013). General Education Curriculum:
Holistic understandings, intellectual and civic competencies
Nocon R. et al. (2018). Essential Mathematics for the Modern World. Quezon City: C & E Publishing,
Inc.
SESSION 1. MATHEMATICS IN OUR WORLD
Good day, learner! With this module, you should be able to
understand how significant mathematics is in our life.
Appreciating mathematics is always one of the easiest thing
for a student to do, but it is also the one most overlooked
notion as to why students cannot answer math problems.
An appreciation of mathematics makes finding the solution
to a problem easier. Mathematics is learned in a lot of
forms, throughout all levels of life. I hope that this module
will provide you an insight and put value in your
mathematical learnings, as well as make a cornerstone for
your future lessons.
What is Mathematics?
Mathematics is defined as the study of numbers and arithmetic operations. Others describe
mathematics as a set of tools or a collection of skills that can be applied to questions of “how many”
or “how much”. Still, others view it is a science which involves logical reasoning, drawing conclusions
from assumed premises, and strategic reasoning based on accepted rules, laws, or probabilities.
Mathematics is also considered an art which studies patterns for predictive purposes or a specialized
language which deals with form, size, and quantity.
Whatever point of view is taken, there is no denying the fact that mathematics is universal. People
from around the world use math to get things done. It helps them perform daily tasks as well as
make important decisions like buying wisely, choosing the most appropriate insurance plan, or even
betting on an outcome with the highest chance of actually occurring. The same mathematical
concepts and languages are applied regardless of whether the users are Asians, Americans, Africans,
or Europeans.
A Study of Patterns
A pattern:
 is an arrangement which helps observers anticipate what they might see or what happens
next.

shows what may have come before.

organizes information so that it becomes more useful. The human mind is programmed to
make sense of data or to bring order where there is disorder.
seeks to discover relationships and connections between seemingly unrelated bits of
information.

Here are examples of pattern – seeking behaviour of humans from childhood to adulthood:

A toddler separates blue blocks from red blocks.


A kindergarten student learns to count.
A first grader does skip counting.

A third grader notices that multiples of two are even numbers.


A sixth grader creates patterns that cover a plane.
A junior high school student learns that a function is essentially a pattern of how one number
is transformed to one another

A college biology undergraduate studies the sequence of DNA and proteins.


A stock trader studies trends in the stock market.
A weatherman makes weather forecasts based on atmospheric patterns.

A doctor decides who is healthy and who is not by recognizing certain health patterns.
Patterns are studied because they are everywhere; people just need to learn to notice them.
Mathematics is the study of patterns. That is one reason why those who use patterns to analyse
and solve problems often find success compared with those who cannot. Understanding new concepts
can also be done in the same way. After all, many results in mathematics come about as
generalizations of patterns in numbers and shapes. Those who recognize, generalize, and use
patterns around them are better at solving problems, have deeper appreciation of the use of
mathematics, and are better equipped to work with mathematics than those who do not.
Studying patterns allows one to observe, hypothesize, discover, and create. Today’s mathematics
is much more than algebra and geometry. The way of doing it has evolved from just performing
calculations or deductions into observing patterns, testing conjectures, and estimating results.
Mathematics has become a diverse discipline that deals with data, measurements, and observations
from science and works with models of natural phenomena, human behaviour, and social systems.
It reveals patterns that help individuals better understand the world and predict what comes next,
imagine what came before, and estimate if the same pattern will occur when variables are changed.
Below are examples of various patterns:

Logic Patterns. Logic patterns are usually the first to be observed. Classifying things, for
example, comes before numeration. Being able to tell which things are blocks and which are
not precede learning to count blocks. One mind of logic pattern deals with the characteristics
of various objects while another deals with order. These patterns are seen on aptitude tests
in which takers are shown a sequence of pictures and asked to select which figures comes
next among several choices.
?
A



B
C
D
Answer: C
Number Patterns. Another class of patterns is the patterns of numbers. Number patterns
such as 2, 4, 6, 8, 10 are familiar to students since they are among the first patterns
encountered in school. Mathematics is especially useful when it helps predict events. “What
will the 10th number of a certain pattern be? “How many coolies would be needed if the party
was for the school instead of just for the class?” Moving on into the higher grades, students
again encounter number patterns through the concept of functions, which is a formal
description of the relationships among different quantities.
8, 13, 18, 23, 28, …
nth term = ?
Geometric Patterns. A geometric pattern is a motif or design that depicts abstract shapes
like lines, polygons, and circles, and typically repeats like a wall paper. Visual patterns are
observed in nature and in art. In art, patterns present objects in a consistent, regular manner.
They appear in paintings, drawings, tapestries, wallpapers, tilings, and carpets. A pattern
does not need to repeat exactly as long as it provides a way of “organizing” the artwork.
Patterns in nature are often more chaotic. Nature provides many examples of patterns,
including symmetries, spirals, tilings, stripes, and fractional dimensions.
Word Patterns. Patterns can also be found in language like the morphological rules on
pluralizing nouns or conjugating verbs for tense, as well as the metrical rules or poetry. Each
of these examples supports mathematical and natural language understanding. The focus
here is patterns in form and in syntax, which lead directly to the study of language in general
and digital communication in particular.
knife : knives
life : lives
wife : ?
mask : masked
walk : walked
balk : ?
Patterns indicate a sense of structure and organization that it seems only humans are capable of
producing these intricate, creative, and amazing formations. It is from this perspective that some
people see an “intelligent design” in the way that nature forms.
a. Snowflakes and Honeycombs
Recall that symmetry indicates that you can draw an imaginary line across an object and the
resulting parts are mirror images of each other.
The figure above is symmetric about the axis indicated by the dotted line. Note that the left
and right portions are exactly the same. This type of symmetry, known as line or bilateral symmetry,
is evident in most animals, including humans. Look in a mirror and see how the left and right sides
of your face closely match.
There are other types of symmetry depending on the number of sides of faces that are
symmetrical.
Note that if the spiderwort and starfish be rotated by several degrees, the same appearance as
the original position will be achieved. This is known as rotational symmetry. The smallest angle that
a figure can be rotated while still preserving the original formation is called the angle of rotation. For
the spiderwort, the angle of rotation is 120° while the angle of rotation for the baby starfish is 72°.
Order of Rotation
1
A figure has a rotational symmetry of order n (n – fold rotational symmetry) if 𝑛 of a complete
turn leaves the figure unchanged. To compute for the angle of rotation, use the following formula:
Angle of rotation =
360°
𝑛
Consider the image of a snowflake.
It can be observed that the patterns on a snowflake repeat six times, indicating that there is a
six – fold symmetry. To determine the angle of rotation, simply divide 360° by 6 to get 60°. Many
combinations and complex shapes of snowflakes may occur, which lead some people to think that
“no two are alike”. Looking closely, however, many snowflakes are not perfectly symmetric due to the
effects of humidity and temperature on the ice crystals as it forms.
Another marvel of nature’s design is the structure and shape of a honeycomb. People have
long wondered how bees, despite their very small size, are able to produce such arrangement while
humans would generally need the use of a ruler and compass to accomplish the same feat. It is
observed that such formation enables the bee colony to maximize their storage using the smallest
amount of wax.
You can try it for yourself. Using several coins of the same size, try to cover as much area of
a piece of paper with coins. If you arrange the coins in a square formation, there are still plenty of
spots that are exposed. Following the hexagonal formation, however, with the second row of coins
snugly fitted between the first rows of coins, noticed that more area will be covered.
Translating this idea to three – dimensional space, we can conclude that hexagonal formations
are more optimal in making use of the available space. These are referred to as packing problems.
Packing problems involve finding the optimum method of filling up a given space such as a cubic or
spherical container. The bees have instinctively found the best solution, evident in the hexagonal
construction of their hives. These geometric patterns are not only simple and beautiful, but also
optimally functional.
b. Tiger’s Stripes and Hyena’s Spots
Patters are also in the external appearances of animals. We are familiar with how a tiger looks
– distinctive reddish – orange fur and dark stripes. Hyena is another predator from Africa, are also
covered in patterns of spots. These seemingly random designs are believed to be governed by
mathematical equations. According to a theory by Alan Turing, the man famous for breaking the
Enigma code during the World War II, chemical reactions and diffusion processes in cells determine
these growth patterns. More recent studies addressed the question of why some species grow vertical
stripes while others have horizontal ones. A new model by Harvard University researchers predicts
that there are three variables that could affect the orientation of these stripes:
 the substance that amplifies the density of stripe patterns;
 the substance that changes one of the parameters involved in stripe formation; and
 the physical change in the direction of the origin of the stripe.
c. The sunflower
Looking at a sunflower up close, noticed that there is a definite pattern of clockwise and
counter clockwise arcs or spirals extending outward from the center of the flower. This is another
demonstration of how nature works to optimize the available space. This arrangement allows the
sunflower seeds to occupy the flower head in a way that maximizes their access to light and necessary
nutrients.
d. The Snail’s Shell
We are very familiar with spiral patterns. The most common spiral patterns can be seen in
whirlpools and in the shells of snails and other similar mollusks. Snails are born with their shells,
called protoconch, which start out as fragile and colorless. Eventually, these original shells harden
as the snails consume calcium. As the snails grow, their shells also expand proportionately so that
they can continue to live inside their shells. This process results in a refined spiral structure that is
even more visible when the shell is sliced. This figure, called an equiangular spiral, follows the rule
that as the distance from the spiral center increases (radius), the amplitudes of the angles formed by
the radii to the point and the tangent to the point remain constant. This is another example of how
nature seems to follow a certain set of rules governed by mathematics.
e. Flower Petals
Flowers are easily considered as things of beauty. Their vibrant colors and fragrant odors
make them very appealing as gifts or decorations. If you look more closely, you will note that different
flowers have different number of petals. Take the iris and trillium, for example, both flowers have
only 3 petals. Flowers with five petals are said to be the most common. These include buttercup,
columbine, and hibiscus. Among these flowers with eight petals are clematis and delphinium, while
ragwort and marigold have thirteen. These numbers are all Fibonacci numbers, which will be
discussed in detail in the later section.
f. World Population
As of 2017, it is estimated that the world population is about 7.6 billion. World leaders,
sociologists, and anthropologists are interested in studying population, including its growth.
Mathematics can be used to model population growth. Recall that the formula for the
exponential growth is A = P𝑒 𝑟𝑡 , where A is the size of the population after it grows, P is the
initial number of people, r is the rate of growth, and t is time. Recall further that e is Euler’s
constant with an approximate value of 2.718. Plugging in values to this formula would
result in the population size after time t with a growth rate of r.
Example: The exponential growth model A = 30𝑒 0.02𝑡 describes the population of the city in
the Philippines in thousands, t years after 1995.
a. What was the population of the city in 1995?
b. What will be the population in 2020?
c. When will the population be 60,000?
Solution
a. Since our exponential growth model describes the population t years after 1995, we
consider 1995 as t = 0 and then solve for A, our population size.
A = P𝑒 𝑟𝑡
standard equation
(0.02)(0)
A = 30𝑒
substitution
A = 30𝑒 0
A = 30(1)
A = 30
final answer
Therefore, the city population in 1995 was 30,000.
b. We need to find A for the year 2020. To find t, we subtract 2020 and 1995 to get t = 25,
which we then plug in to our exponential growth model.
A = P𝑒 𝑟𝑡
standard equation
(0.02)(25)
A = 30𝑒
substitution
A = 30𝑒 0.5
A = 30(1.648721271)
A = 49.4616
final answer
Therefore, the city population would be about 49,462 in 2020.
c. To find the year when 𝐴 = 60, we need to solve for t first and then add it to 1995.
A = P𝑒 𝑟𝑡
standard equation
𝐴
transpose P to left side and use symmetric property of equality
𝑒 𝑟𝑡 = 𝑃
𝐴
𝑟𝑡 = ln (𝑃)
𝑡=
𝑡=
𝐴
𝑃
ln ( )
𝑟
60
30
ln ( )
0.02
ln 2
use natural logarithm to solve for the exponent of e
final equation
substitution
𝑡 = 0.02
evaluate
estimate
2
1995 + 34.66 = 2029.66 𝑜𝑟 2029 𝑜𝑟 2029 𝑎𝑛𝑑 𝑜𝑛 𝑡ℎ𝑒 8𝑡ℎ 𝑚𝑜𝑛𝑡ℎ
3
𝑡 = 34.65735903 …
𝑡 = 34.66
NOTE: In solving mathematical problems, provide the given, basic and final equation (some problems
have the same basic and final equation, others do not), substitution, evaluation and final
answer.